Download Department of Biomedical Engineering

Department of Biomedical Engineering
Investigation of transmural cardiac and
fiber strain in ischemic and non-ischemic
tissue during diastole
Katarina Lundgren
LITH-IMT/BMS20-EX--06/438--SE
Linköping 2006
Department of Biomedical Engineering
Linköpings universitet
SE-581 85 Linköping, Sweden
Linköpings tekniska högskola
Linköpings universitet
581 85 Linköping
Investigation of transmural cardiac and
fiber strain in ischemic and non-ischemic
tissue during diastole
Master’s thesis
performed at Biomedical Modelling and Simulation,
Department of Biomedical Engineering
at Linköpings universitet
by Katarina Lundgren
LITH-IMT/BMS20-EX--06/438--SE
Supervisor:
Katarina Kindberg
Dept. of Biomedical Engineering, Linköpings universitet
Examiner:
Matts Karlsson
Dept. of Biomedical Engineering, Linköpings universitet
Linköping, 14 December, 2006
Avdelning, Institution
Division, Department
Datum
Date
Division of Biomedical Modelling and Simulation
Department of Biomedical Engineering
Linköpings universitet
S-581 83 Linköping, Sweden
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Language
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Report category
ISBN
Svenska/Swedish
Licentiatavhandling
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Övrig rapport
2006-12-14
—
LITH-IMT/BMS20-EX--06/438--SE
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URL för elektronisk version
http://www.imt.liu.se
www.diva-portal.org/liu/undergraduate/
Titel
Title
Investigation of transmural cardiac and fiber strain in ischemic and non-ischemic
tissue during diastole
Undersökning av transmurala hjärt- och fibertöjningar i ischemisk och ickeischemisk vävnad under diastole
Författare Katarina Lundgren
Author
Sammanfattning
Abstract
The cardiac wall has complex three-dimensional fiber structures and mechanical
properties that enable the heart to efficiently pump the blood through the body.
By studying the myocardial strains induced during diastole, information about the
pumping performance of the heart and what mechanisms that are responsible for
this effective blood filling, can be achieved. Two different computation methods
for myocardial strain, both based on data acquired from marker technique, were
compared using a theoretical cylinder model. The non-homogeneous polynomial
fitting method yielded higher accuracy than a homogeneous tetrahedron method,
and was further used to investigate cardiac and fiber strains at different wall depths
and myocardial regions in normal and ischemic ovine hearts. Large spatial and
regional variations were found, as well as alterations, conveyed by ischemic conditions, of fiber mechanisms responsible for the circumferential expansion and wall
thinning during diastole.
Nyckelord
Keywords
strain, cardiac, fiber, transmural, ischemic, tensor, markers
Abstract
The cardiac wall has complex three-dimensional fiber structures and
mechanical properties that enable the heart to efficiently pump the
blood through the body. By studying the myocardial strains induced
during diastole, information about the pumping performance of the
heart and what mechanisms that are responsible for this effective
blood filling, can be achieved. Two different computation methods
for myocardial strain, both based on data acquired from marker technique, were compared using a theoretical cylinder model. The nonhomogeneous polynomial fitting method yielded higher accuracy than
a homogeneous tetrahedron method, and was further used to investigate cardiac and fiber strains at different wall depths and myocardial regions in normal and ischemic ovine hearts. Large spatial and
regional variations were found, as well as alterations, conveyed by
ischemic conditions, of fiber mechanisms responsible for the circumferential expansion and wall thinning during diastole.
v
Acknowledgments
First of all I would like to thank my supervisor Katarina Kindberg
who has been an excellent and very competent tutor during my thesis
work. She and Charlotte Oom, who is a good friend and former colleague at IMT, have always been available for discussion and support,
which has been invaluable in my work. I would also like to thank my
examinator Matts Karlsson for inspiring me and making this project
possible. All other colleagues at IMT deserve acknowledgments as
well, for creating a warm and joyful everyday environment. Finally,
my dear friends and beloved family have, as always, been a great encouragement and support to me.
Katarina Lundgren
Linköping, December 2006
vii
Contents
1 Introduction
1.1 Aims of this Master’s Thesis . . . . . . . . . . . . . . .
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2 Cardiac Anatomy, Physiology and
2.1 Anatomy and Physiology . . . . .
2.1.1 The Circulation System .
2.1.2 The Cardiac Wall . . . . .
2.1.3 The Cardiac Cycle . . . .
2.2 Myocardial Ischemia . . . . . . .
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3 Cardiac Kinematics
3.1 Strain . . . . . . .
3.2 Tensors . . . . . .
3.3 The Strain Tensor .
3.4 Strain Components
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Pathology
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4 Data Acquisition
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4.1 Marker Tracking . . . . . . . . . . . . . . . . . . . . . 17
4.2 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Comparison of Strain Computation Methods
5.1 Cylinder Model . . . . . . . . . . . . . . . . .
5.2 Homogeneous Tetrahedron Method . . . . . .
5.3 Non-Homogeneous Polynomial Fitting Method
5.4 Error Calculation . . . . . . . . . . . . . . . .
5.5 Results of the Comparison . . . . . . . . . . .
5.6 Conclusion of the Comparison . . . . . . . . .
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6 Method
6.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . .
6.1.1 Cardiac Coordinate System . . . . . . . . . . .
6.1.2 Fiber Coordinate System . . . . . . . . . . . . .
6.1.3 Transformation . . . . . . . . . . . . . . . . . .
6.2 Cardiac Cycle Timing . . . . . . . . . . . . . . . . . .
6.3 Cardiac Strain Computation . . . . . . . . . . . . . . .
6.4 Fiber Strain Computation . . . . . . . . . . . . . . . .
6.5 Fiber Strain Contributions to Cardiac Strain Components
6.6 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1 Mean Values and Standard Errors . . . . . . . .
6.6.2 Significance Tests . . . . . . . . . . . . . . . . .
6.7 Missing Markers . . . . . . . . . . . . . . . . . . . . . .
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7 Results
7.1 Baseline . . . . . . . . . . . . . . . .
7.1.1 Cardiac Strains . . . . . . . .
7.1.2 Fiber Strains . . . . . . . . .
7.1.3 Fiber Strain Contributions to
and Radial Strain . . . . . . .
7.2 Ischemic . . . . . . . . . . . . . . . .
7.2.1 Cardiac Strains . . . . . . . .
7.2.2 Fiber Strains . . . . . . . . .
7.2.3 Fiber Strain Contributions to
and Radial Strain . . . . . . .
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8 Discussion
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8.1 Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.2 Ischemic . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 57
Bibliography
59
A Tables
63
Chapter 1
Introduction
The pumping of the heart is created when the cardiac muscle fibers
actively contract and relax. These course of events give rise to large
elastic deformations in the cardiac wall during each cardiac cycle. The
mechanical properties of fibers, in particular their specific architecture,
affect the pumping performance of the ventricles. Thus, by investigating myocardial strain, greater knowledge about myocardial function is
achieved.
1.1
Aims of this Master’s Thesis
The purpose of this thesis work is to, by the use of surgically implanted radiopaque markers, investigate myocardial strain. The work
is divided into two parts, where the first part describes and evaluates
two different methods of computing transmural strain. In the second
part one of them used is to perform strain calculations for real ovine
data sets. The strain is characterized in local cardiac coordinates and
in a fiber coordinate system. It is further analyzed for two diastolic
time points, at two different myocardial regions at three wall depths.
The procedure is performed for data derived from sheep in normal
conditions and ischemic condition, regarded as two separated cases.
1
2
Introduction
Chapter 2
Cardiac Anatomy,
Physiology and Pathology
In this chapter the basic anatomy and physiology of the heart is described, with focus on the cardiac wall, its structure and its function
regarding the pumping effect of the heart. Also a brief review on the
condition of ischemia is given.
2.1
2.1.1
Anatomy and Physiology
The Circulation System
The circulation system of the blood has its center in the heart, where it
receives the energy needed to reach all parts of the body. Oxygen depleted blood returns to the heart via three major veins; superior vena
cava, inferior vena cava and coronary sinus and enters the heart in the
right atrium. Inferior of the atrium is the right ventricle, forming most
of the anterior surface of the heart, and between the chambers there is
a tricuspid valve. When the pulmonary valve opens blood is pumped
out from the right ventricle into the pulmonary arteries leading to the
lungs. Oxygenated blood leaves the lungs via the pulmonary veins
and reenters the heart in the left atrium. Between the left atrium and
the left ventricle there is a bicuspid valve called the mitral valve. Both
the bicuspid and the tricuspid valves are anchored by tendonlike cords
connected to papillary muscles shaped like cones and attached to the
endocardium. These prevent the valves from being inverted and limit
3
4
Cardiac Anatomy, Physiology and Pathology
movements of the valves. The left ventricle pumps the blood via the
aortic valve into the aorta and further out to all body tissues. The
pointed end of the left ventricle forms the apex of the heart, and is
directed anteriorly and to the left. The wall separating the two ventricles is called the septum. [20]
In order to supply the heart efficiently enough with oxygen and nu-
Figure 2.1. Anatomy of the heart
trients the heart has its own blood circulation system, the coronary
circulation, that swirls and branches on the epicardium of the heart
as arteries and veins. Each part of the myocardium has contact with
more than one coronary artery, guaranteeing sufficient oxygen even if
obstruction of one artery occurs. Blood can flow through the coronary
vessels during relaxation, and while the heart muscles are contracting
the coronary vessels are squeezed shut and little blood can pass. [20]
2.1 Anatomy and Physiology
2.1.2
5
The Cardiac Wall
The walls of the chambers in the heart can be divided into three distinct layers: the outermost epicardium, the middle myocardium and
the innermost endocardium. The heart is surrounded by a fluid filled
sac called the pericardium which keeps the heart in place, prevents
over expansion during blood filling and limits heart motions. The
inner part of the pericardium, closest to the heart’s surface, is the
protective epicardium layer, composed of connective tissue (collagen
and some elastic fibers) and epithelial tissue. The inside of the heart,
including the valves and the inner lining of the large blood vessels
connected to the heart, is covered with the endocardium layer. Also
this layer consists of connective and epithelial tissue. [9, 20]
The myocardium layer is the active part of the cardiac walls, thus
here are the contractile cardiac muscle tissue situated. This tissue is a
unique type of muscle tissue, where the muscle fibers are built up by irregular shaped, mononuclear cells called myocytes. The relative short
cardiac muscle fibers are connected end-to-end by intercalated discs
containing gap junctions where conduction of muscle action potential
takes place [20]. The fibers are also connected parallely side-to-side,
making up laminar sheets of three to four cells thick. Within the
sheets the myocytes are tightly coupled by a network of extracellular
collagen fibers, but between adjacent sheets the myocyte coupling is
looser [14]. These fiber sheets have at any point in a normal heart
a clear predominant fiber direction, aligning the plane of the sheet
and being approximately tangent with the cardiac wall [17]. The fiber
direction is different depending on the location in the cardiac muscle.
Regarding the left ventricular free wall there is a denser packing of
myocytes and more intramyocardial collagen in the outer half of the
wall than in the inner half [9].
The myocardial fibers form helices that have a left-handed orientation
at the epicardium and a right-handed orientation at the endocardium.
The pitch of the helices, how much they differ from the circumferential direction, is described by the angle α. Typical α-values have been
measured to vary from α ≈ −60◦ at the epicardium, to α ≈ 60◦ at
the endocardium for several species [8]. There is also a sheet angle, β,
6
Cardiac Anatomy, Physiology and Pathology
Figure 2.2. The fiber and sheet structure at three transmural levels of
the ovine lateral equatorial left ventricle. Cardiac coordinates are circumferential XC , longitudinal XL and radial XR , and fiber Xf , sheet Xs and
sheet-normal Xn are axes in the fiber coordinate system.
that represents how much the myocardial sheets are rotated around
the fiber direction relative to the radial direction. In the ovine lateral
region near the epicardium and near the endocardium the sheets are
typically tilted with a sheet angle of 45◦ , but in the midwall β ≈ −45◦ .
For the basal anterior region this pattern is the opposite [8]. Mean
fiber and sheet angles of the sheep used in this study can be found in
table 4.1.
The principle shape and dimensions of the heart vary with species,
age, phase of the cardiac cycle and disease. In particular the thickness
of the ventricular walls vary regionally and temporally. The wall of
the left ventricle is thicker than that of the right ventricle, because
the ejection from the left ventricle needs to be more powerful in order
to pump blood a greater distance. In general for a normal heart, the
2.2 Myocardial Ischemia
7
ventricular walls are thickest at the equator and base of the left ventricle and thinnest at the left ventricular apex and right ventricular
free wall. [17]
2.1.3
The Cardiac Cycle
The phase in the cardiac cycle when the ventricles contract and the
blood is pumped out of the heart, is called (ventricular) systole. The
volume is maximum and when the pressure in the left ventricle exceeds that of the aorta, the aortic valve opens and blood is ejected.
There is a time of isovolumetric contraction, before blood ejection,
when all four valves are closed and the muscle fibers contract without
yet shortening. The myocytes in the ventricular walls then start to
relax and repolarize, resulting in a drop of ventricular pressures. Here
follows an isovolumetric relaxation period when again all four valves
are closed and the volumes are constant. When left ventricular pressure reaches below the pressure of the atrium the mitral valves open
and ventricular blood filling begins. This is considered as the start of
diastole which is also indicated in the pressure-volume loop in figure
2.3. The filling phase ends when the ventricular volumes have reached
their maximum and the mitral and tricuspid valves close and systole is
about to take place again, end diastole (ED). The valves in the heart
also prevent the blood from flowing backwards. [20]
2.2
Myocardial Ischemia
The most common cause of heart diseases and in many western countries the number one cause of death is coronary heart diseases causing myocardial ischemia [18]. The majority of these patients suffer
from atherosclerosis, where the coronary arteries are obstructed by
atherosclerotic plaques [18]. Why the plaques are formed is not known
in detail, but there are many probable factors that can cause damage
to the endothelial lining of artery walls and thereby initiating the aggregation of platelets and attracting phagocytes. This inflammation
and further clotting of cholesterol, lipids, arterial smooth muscle cells
and their products result in a blockage of blood flow in that coronary
artery and diminishing the oxygen supply to the heart [20]. Moreover there is always a risk that a piece of the plaque can dislodge
8
Cardiac Anatomy, Physiology and Pathology
Figure 2.3. Pressure-volume loop of left ventricle for three consecutive
representative heart beats. End diastole (ED).
and obstruct blood flow in other arteries. Some of the risk factors
of atherosclerosis are high level of the cholesterol carrier LDL (lowdensity lipoprotein), prolonged high blood pressure, carbon monoxide
in cigarette smoke and high blood glucose levels in diabetes [10]. Also
blood clots, produced at thrombosis, when blood coagulates too easily,
and tumors can be reasons of artery obstruction [20].
A condition of myocardial ischemia can appear if obstructions in the
coronary arteries cause such a reduction of blood flow that the current oxygen requirement in the myocardium cannot be fulfilled. The
myocardial cells are weakened but not killed in the resulting hypoxia
state, and the symptoms of ischemia can be none (silent myocardial
ischemia) or angina pectoris meaning “strangled chest” which often
appears during exercise when the oxygen demand is higher [10]. If the
blood flow in the artery is totally obstructed the outcome may be a
myocardial infarction, meaning an anoxic state causing necrosis (cell
death). The affected area of the myocardium become dead noncontractile tissue [20]. The consequences on the heart function depend on
the size and location of the infarcted area. In a hypoxia state the damaged cells can still be able to conduct the electrical impulses, though
at a slower rate, but in a severe infarction the affected area can cause
2.2 Myocardial Ischemia
9
disruptions in the conduction system triggering ventricular fibrillation
[10, 20].
In an attempt to maintain the functions of a failing heart, the heart
often uses compensation mechanisms. It can be a dilation of the cardiac muscle fibers in order to achieve greater power, hypertrophy of
the ventricles which means an increase in organ size, or a higher pulse
frequency. At first the compensations have positive effects, but after
time they mean more stress for the heart and via regulation mechanisms also negative effects on other organs, for example the kidneys
[18].
10
Cardiac Anatomy, Physiology and Pathology
Chapter 3
Cardiac Kinematics
To fully describe the pumping performance of the heart, cardiac pressures and volumes are basic and good measures but not always sufficient. Myocardial strains are used to characterize ventricular function,
especially in investigations regarding the cardiac fiber architecture,
and the effect of pathological conditions like ischemia, as done in this
thesis.
3.1
Strain
Deformations can occur when a solid body is subjected to forces. The
deformation is the change in distance between any two points of the
body from the reference unloaded state to the deformed state when
forces are applied. Translation and rotation can appear even though
the distance remains and thus no deformations occur. The deformations in the cardiac wall are elastic, meaning that once the forces are
no longer applied the body returns to its former reference state.
Strain describes deformation as a dimensionless relative ratio. Lagrangian infinitesmal strain in one dimension is the ratio between the
lengthening (or shortening) of the distance between two points along
this direction and the distance in the reference state. For deformations in more than one dimension there are not only normal strains
along each direction but also shear strains, see section 3.4 for further
specification.
11
12
3.2
Cardiac Kinematics
Tensors
A tensor is a mathematical tool often used to represent and analyze
physical states, such as strains. The tensor as a tool is independent of
which coordinate system it is calculated and presented in. However,
the content of the tensor (the strain components) is dependent on
the choice of coordinate systems. A tensor of order zero is a scalar
and has one single component, a first order tensor is a vector and has
three components in 3D. Three dimensional strain is represented by a
second order tensor which is a matrix and has nine scalar components.
[7]
3.3
The Strain Tensor
Defining a deformation with a strain tensor is possible by looking at
the change of the distance between two points and using the dependence between the coordinates in the different states.
Let (X1 , X2 , X3 ) be the coordinates of a point P of a body in a reference configuration as seen in figure 3.1. A neighboring point P 0 has the
coordinates (X1 + dX1 , X2 + dX2 , X3 + dX3 ) and is connected with P
by the infinitesimal line element P P 0 . The squared distance between
these points is the squared length of P P 0 given by
ds20 = dX12 + dX22 + dX32
(3.1)
When a deformation takes place P and P 0 transform to the points p
and p0 with the coordinates (x1 , x2 , x3 ) and (x1 + dx1 , x2 + dx2 , x3 + dx3 )
respectively, in a new coordinate system. The squared distance between the points in this deformed configuration is then
ds2 = dx21 + dx22 + dx23
(3.2)
The dependence between the coordinate systems can be expressed as
xi = xi (X1 , X2 , X3 ),
(3.3)
saying that the deformation of the body can be known if coordinates
in the deformed state are known functions of the coordinates in the
3.3 The Strain Tensor
13
Figure 3.1. Displacement and deformation of a body.
reference state.
From equation (3.3) it follows that the infinitesimal line components
in the deformed coordinate system can be expressed as a summation
of partial derivatives, and written with the Einstein summation convention as
∂xi
dxi =
dXj
(3.4)
∂Xj
Introducing the Kronecker delta
1 if i = j
δij =
0 if i 6= j
(3.5)
the squared distances can be written as
ds20 = δij dXi dXj
(3.6)
ds2 = δαβ dxα dxβ
(3.7)
14
Cardiac Kinematics
and by using (3.4) and changing the indexes, (3.7) can be further
developed into
∂xα ∂xβ
dXi dXj
(3.8)
ds2 = δαβ
∂Xi ∂Xj
The difference between ds2 and ds20 can now be written as
∂xα ∂xβ
2
2
ds − ds0 = δαβ
− δij dXi dXj
∂Xi ∂Xj
(3.9)
From definition, the Langrangian strain tensor in indicial notation
is
∂xα ∂xβ
1
δαβ
− δij
(3.10)
Eij =
2
∂Xi ∂Xj
thus
ds2 − ds20 = 2Eij dXi dXj
(3.11)
and thus the deformation of the body is expressed with a strain tensor
E [7].
In matrix notation E is defined by
1
1
E = (C − I) = (F T F − I)
2
2
(3.12)
where C is the Cauchy-Green deformation tensor, I is the identity
matrix and F is the deformation gradient tensor defined as
 ∂x1 ∂x1 ∂x1 
∂X1
∂X2
∂X3
∂x
∂x2
∂x2
∂x2 
=  ∂X
F =
(3.13)
∂X2
∂X3
1
∂X
∂x3
∂x3
∂x3
∂X1
∂X2
∂X3
C is derived from the inner product of the tensor F and its transpose,
which makes C symmetric. Hence E is also symmetric, since I being
diagonal.


E11 E12 E13
E =  E21 E22 E23 
(3.14)
E31 E32 E33
E12 = E21 , E13 = E31 and E23 = E32 .
3.4 Strain Components
15
When the Cauchy-Green deformation tensor C in (3.12) equals the
identity matrix I, the strain tensor E turns zero. Meaning that there
has been no change of the distances from one time frame to another,
ds2 − ds20 = 0, thus no deformation. This would be the case for a rigid
body motion (translation and/or rotation) or in absence of motion,
neither of the cases causes any deformation of the body [19].
3.4
Strain Components
The nine scalar components mentioned in section 3.2 are seen in (3.14),
where six of them are independent components as three normal strains
and three shear strains. The diagonal elements E11 , E22 and E33
are normal strains along respective coordinate axis, and describe an
expansion or dilation in each direction. These strains will deform a
cube into a beam. E12 , E13 and E23 are shear strains, representing
angle changes between two of the intially orthogonal coordinate axes.
A cube will be deformed into a parallelepiped by the shear strains [2].
16
Cardiac Kinematics
Chapter 4
Data Acquisition
The data analyzed in this study is acquired from an invasive marker
tracking method, a method extensively used for cardiac kinematics research, yielding high spatio-temporal resolution of the tracked points.
Here, the basics of the acquisition technique and the character of the
acquired data sets are explained.
4.1
Marker Tracking
Radiopaque markers have surgically been implanted in ovine (sheep)
hearts at Falk Cardiovascular Research Center, Stanford University
School of Medicine, Stanford, CA, USA. The ovine heart was chosen
to study because of its anatomical similarities to the human heart and
its highly consistent, reproducible anatomy, function and patterns of
dysfunction [11]. A detailed description of the surgical preparation
and operation, and the anesthesia of the sheep can be found in [4].
13 markers were inserted in the subepicardium of the left ventricle to
silhouette the chamber, as illustrated in figure 4.1(a). Further, transmural bead arrays of 12 markers were inserted; at a lateral-equatorial
position below marker 12, between the papillary muscles of the left
ventricle [4], and/or at a anterior region, basal to the anterior papillary muscles, above marker 4 [3]. Figure 4.1(b) shows the three transmural columns of four marker beads each, normal to the epicardial
tangent plane, that make up a bead array with the marker number
15-26. Marker 15, 19 and 23 are on the epicardial surface.
17
18
Data Acquisition
(a)
(b)
Figure 4.1. Schematic pictures over inserted markers in the left ventricle.
(a) Silhouetting markers 1-13 and transmural bead arrays at lateral and/or
anterior space. (b) A close-up of a bead array with three columns of four
markers each, numbers: 15-18, 19-22, 23-26, where markers 15,19 and 23
are placed on the epicardial surface. Cardiac coordinate axes indicated as
XC - circumferential, XL - longitudinal and XR - radial.
After the bead implantation the chest was closed and the sheep allowed to recover for eight weeks. Then the data acquisition with biplane videofluoroscopic imaging of the radiopaque markers was carried
out at 60 Hz. By taking two-dimensional images from two different
angles, three-dimensional coordinates of the markers can be obtained.
Additional information as left ventricular pressure (LVP) and surface
lead ECG was acquired and saved on the same time. All measurements were taken during three consecutive heart beats with the heart
in sinus rhythm and steady-state normal conditions. As far as possible
all three heart beats have been used in the analysis in this study.
The ovine hearts were arrested after the data acquisition when the
end diastolic pressure was attained. The fiber angle α and sheet angle
β (explained in chapter 2.1.2) were determined using quantitative histology, which is thoroughly described in [5]. The blocks of myocardial
tissue examined were taken adjacent and basal to the concerned implanted bead arrays, and the angles were measured at three different
4.2 Data Sets
19
transmural depths. The measured mean angles for the data sets are
presented in table 4.1.
4.2
Data Sets
The marker tracking method was performed on 15 healthy adult sheep,
from which baseline data sets were derived. Ischemic data sets were
acquired from ten additional sheep. These ischemic sheep got snares
around one or two obtuse marginal branches of the left posterior coronary artery inserted, but not tightened, at the same intervention as
the marker insertion. After a one week recovery period, the inserted
coronary artery snares were tightened, producing a complete occlusion
of the selected vessels to provoke the myocardial ischemic condition.
All the sheep were allowed to recover for eight weeks from the marker
insertion, before the data acquisition took place, and no further interventions occurred during this time. Bead arrays were inserted at
lateral and/or at anterior positions. Ten of the baseline animals had
lateral bead arrays and seven of them anterior arrays. For the ischemic
sheep nine animals received lateral bead arrays and six got bead arrays at the anterior site. A table specifying performed interventions
for each animal can be found in the appendix table A.1.
Table 4.1. Mean angle values ± standard deviation (SD) for all available
animals within specified data set. Baseline (B) and ischemic (I).
α[◦ ]
Data set
B lat (N=7)
B ant (N=6)
I lat (N=8)
I ant (N=6)
subepi
−37±8
−29±9
−15 ±10
−26 ±10
mid
−9 ±9
−5 ±7
−7 ±16
3 ±6
β[◦ ]
subendo
18 ±10
17 ±17
12 ±21
33 ±11
subepi
37 ±33
−8 ±49
12 ±40
−14 ±44
mid
−37 ±6
40 ±11
−43 ±25
19 ±44
subendo
62 ±12
−30 ±34
−15 ±48
−27 ±45
20
Data Acquisition
Chapter 5
Comparison of Strain
Computation Methods
In this chapter two methods for computing cardiac strain will be evaluated. The evaluation will be done by comparing the strain that each
method delivers to an analytical strain obtained from synthetic data of
a cylinder. From this, strain errors can be computed and the accuracy
of each method determined.
5.1
Cylinder Model
For the validation of the methods described in the subsequent sections,
a synthetic data set from a cylinder model is used. This model is
created by deforming a cylinder with known deformation parameters.
The cylinder has the cylindrical coordinates (R, Θ, Z) in the reference
configuration with an inner radius of R1 = 2 cm and outer radius R2 =
3 cm, and coordinates (r, θ, z) in the deformed state. The deformation
is then described by the following relations:
q
R2 −R12
+ r12
r=
λ
(5.1)
θ = φR + Θ + βZ
z = ωR + λZ
where the kinematic parameters are chosen to φ = 0.1 rad cm−1 and
ω = 0.3 for transverse shear, β = 0.2 rad cm−1 for torsion, λ = 0.8
for axial extension ratio and the deformed inner radius r1 = 1.65 cm.
21
22
Comparison of Strain Computation Methods
Since the methods are to be applied on real data sets extracted from
cardiac walls, the cylinder is meant to be a model of the heart and the
parameters are chosen so that the deformation resembles that of the
left ventricle during contraction. [16]
Figure 5.1. Cylindrical model of the left ventricle in an undeformed state
to the left and a deformed state to the right. The deformation is known
and given by the equation 5.1
The cylinder can also be described in a cartesian coordinate system
with coordinates (X1 , X2 , X3 ) in the reference state and (x1 , x2 , x3 ) in
the deformed state. The transformation from the cylindrical system
looks like
X1 = R sin Θ
X2 = Z
(5.2)
X3 = R cos Θ
Thus, in the cartesian coordinate system X1 is circumferential, X2 is
longitudinal and X3 is radial.
In order to resemble the real ovine data sets acquired from the bead
array described in 4.2, three transmural columns of four beads each are
simulated in the cylinder. The three columns make up an isosceles triangle, seen from the outer or the inner surface of the cylinder, were the
base is 1.04 cm and the sides 0.96 cm on the epicardium. The beads
5.2 Homogeneous Tetrahedron Method
23
are equally spaced between the endocardium and the epicardium, as
in figure 5.2(a). The coordinates of the beads are sampled before and
after the deformation, and used to compute both the analytical strain
and the estimated strains.
A deformation gradient tensor can be defined for cylindrical coordinates, as F rθz in the equation below [19]. The analytical strain E ana
is then obtained via the F rθz , and after rotation expressed in cartesian
coordinates according to
 ∂r 1 ∂r ∂r 
∂R
F x1x2x3

 ∂θ
T
T 
= R F rθz R0 = R  r ∂R

∂z
∂R
R ∂Θ
∂Z
r ∂θ
R ∂Θ
∂θ
r ∂Z
1 ∂z
R ∂Θ
∂z
∂Z


 R0


(5.3)
RT and R0 are rotation matrices converting the deformed and undeformed states, respectively, into cartesian coordinates [19];


sin θ cos θ 0
0
1 
RT =  0
cos θ − sin θ 0
(5.4)


sin Θ 0 cos Θ

R0 = cos Θ 0 − sin Θ 
0
1
0
5.2
Homogeneous Tetrahedron Method
Bead coordinates of four non-coplanar points making up the corners of
a tetrahedron are needed in order to calculate strain with this method.
Between these points six line segments can be drawn, as done in figure 5.2(b). By analyzing the change of the length of infinitesimal line
segments that a deformation causes, the strain can be computed, according to section 3.3. The tetrahedrons create finite line segments
between its corners which are analyzed with an approximated finite
version of this method. Further, the strain obtained is assumed to be
constant and homogeneous within the tetrahedron volume. For these
reasons it is relevant to keep the tetrahedrons adequate small.
24
Comparison of Strain Computation Methods
(a)
(b)
Figure 5.2. (a)Tetrahedrons in the bead array. (b) Line segments formed
from one tetrahedron.
When equation (3.11) is applied on the six finite independent line
segments (∆s01 ,∆s02 ,.., ∆s06 in reference state) six linear algebraic
equations are obtained. The six independent strain components are
found by solving these equations. After summation development, the
equation system can be written in a matrix notation as
3 2
2 2
∆s1 − ∆s201
(2∆X12 )1
...
)2
7 6(
6
7 6
6
···
)3
7 6(
6
=
7 6
6
···
)4
7 6(
6
5 4(
4
···
)5
(
)6
∆s26 − ∆s206
(4∆X1 ∆X2 )1
(
)2
(
)3
(
)4
(
)5
(
)6
(4∆X1 ∆X3 )1
(2∆X22 )1
···
···
···
···
···
(4∆X2 ∆X3 )1
32
3
(2∆X32 )1 E11
(
)2 76E127
76
7
(
)3 76E137
76
7
(
)4 76E227
(
)5 54E235
(
)6 E33
(5.5)
⇔ s = Ke
An inversion then gives the six independent estimated strain components in e:
e = K −1 s
(5.6)
The d’s in equation (3.11) have been replaced by ∆’s as a remark of
the approximation into finite length elements in the method.
Figure 5.2(a) illustrates the simulated 3-by-4 bead array in the cylinder. It contains four transmural beads per column, from where three
transmural tetrahedrons can be created, and hence three sets of strain
5.3 Non-Homogeneous Polynomial Fitting Method
25
at different wall depths can be computed. The tetrahedrons can be
chosen in different ways. They can be endocardial-oriented (as in figure 5.2(a)) or epicardial-oriented, meaning that the gravity point of
the tetrahedrons are either closest to the endocardium or to the epicardium. Further, for each one of these ways the fourth corner of the
tetrahedron can be chosen in three different ways (as a bead in column
1, column 2 or in column 3). So for this case with a 3-by-4 bead array
there are six different kinds of the tetrahedron methods available.
5.3
Non-Homogeneous Polynomial Fitting
Method
A coordinate xi in the deformed coordinate system can be expressed as
a function of the coordinates X1 , X2 and X3 in the reference system.
In the non-homogeneous polynomial fitting method a polynomial function is used to estimate the coordinates of a bead j in the deformed
state:
x̂ij = x̂ij (X1j , X2j , X3j )
(5.7)
An estimate of the deformation gradient tensor F̂ in equation (3.13)
can be calculated by differentiation of the polynomial function, and
thus also an estimate of the Lagrangian strain tensor;
1 T
Ê = (F̂ F̂ − I)
2
(5.8)
The polynomial function is obtained by fitting polynomials of different
orders along the reference coordinate directions X1 , X2 and X3 . To
fit a polynomial of order n in direction i there have to be n + 1 beads
along that direction, in order to get a soluble equation system. Since
there for this method are 4 transmural beads simulated in each column in the cylinder, it would be possible to fit a polynomial of first,
second or third order in the radial direction X3 . Here, a quadratic
polynomial fi with three terms will be fitted:
fi (X3 ) = a1i X32 + a2i X3 + a3i
(5.9)
26
Comparison of Strain Computation Methods
Along a (X1 , X2 )-plane there are three beads, thus only polynomials
of first order can be fitted in the X1 and X2 directions and no bilinear
term is possible. This results in a linear polynomial gi of two variables:
gi (X1 , X2 ) = b1i X1 + b2i X2 + b3i
(5.10)
Multiplying (5.9) and (5.10) gives the linear-quadratic polynomial
used to estimate the coordinate xi :
x̂i = fi (X3 )gi (X1 , X2 ) = (a1i X32 + a2i X3 + a3i )(b1i X1 + b2i X2+ b3i) =
c1i
 . 
2
2
2
= X1 X3 X1 X3 X1 X2 X3 X2 X3 X2 X3 X3 1  .. 
c9i
(5.11)
This polynomial is further used to estimate coordinates of all n beads.
For the cylinder model there is a total of twelve beads (n = 12) and
thus twelve equations can be arranged for each coordinate direction
according to the following equation system:


x̂i1


x̂i =  ...  =
x̂in

2
X11 X31

..
=
.
2
X1n X3n
2
2
X11 X31 X11 X21 X31
X21 X31 X21 X31
X31
..
..
..
..
..
..
..
.
.
.
.
.
.
.
2
2
X3n
X2n X3n X2n X3n
X1n X3n X1n X2n X3n
 
1 c1i
..  .. 
.  . 
1
⇔ x̂i = M ci
(5.12)
The sum of the squared differences between the true deformed bead
coordinates (xi1 , ..., xin ) and the estimated deformed bead coordinates
(x̂i1 , ..., x̂in ),
n
X
(xij − x̂ij )2 ,
(5.13)
j=1
is minimized by a least-squares approximation;
ci = (M T M )−1 M T xi
(5.14)
c9i
5.4 Error Calculation
27
This is done for each coordinate direction so that three sets of constants are found (c1 , c2 and c3 ).
∂x
of the polynomial function (5.11) gives the
The differentiation ∂X
following estimation of the components of the deformation gradient
tensor:
c11 X32 + c21 X3 + c31
c41 X32 + c51 X3 + c61
c11 X1 X3 + c21 X1 + 2c41 X2 X3 + c51 X2 + 2c71 X3 + c8
c12 X32 + c22 X3 + c32
c42 X32 + c52 X3 + c62
2c12 X1 X3 + c22 X1 + 2c42 X2 X3 + c52 X2 + 2c72 X3 + c8
c13 X32 + c23 X3 + c33
c43 X32 + c53 X3 + c63
2c13 X1 X3 + c23 X1 + 2c43 X2 X3 + c53 X2 + 2c73 X3 + c8
(5.15)
Hence, from (5.15) and (5.8) a strain estimate can be computed for
an arbitrary point with reference coordinates (X1 , X2 , X3 ) within the
area of the beads. For the cylinder model this is done for 101 points
along a radial line in the gravity center of the triangles formed by the
three bead columns.
F̂ 11
F̂ 12
F̂ 13
F̂ 21
F̂ 22
F̂ 23
F̂ 31
F̂ 32
F̂ 33
5.4
=
=
=
=
=
=
=
=
=
Error Calculation
The accuracy of each method is determined here by calculating the
absolute error for each strain component, according to
(Eij =| Eˆij − Eij |,
(5.16)
where Êij and Eij are the respective strain components from estimated and analytical methods.
The analytical strain and the error calculation are for each method
computed at the same points as the corresponding method yields its
estimated strain. For the tetrahedron method this means three transmural points, situated at the gravity points of the chosen tetrahedrons.
And for the polynomial method, the strain points are 101 points along
28
Comparison of Strain Computation Methods
a radial gravity line. The absolute errors are then integrated across
the radius. The integration is performed for the whole defined cylinder
radius, meaning that the first and last integration interval, in the case
for the tetrahedron method, have different lengths depending on how
the tetrahedrons have been chosen. But the intervals always start and
end with the cylinder radius.
5.5
Results of the Comparison
Resulting analytical and estimated strain components are plotted against
the cylinder radius in figure 5.3. How the tetrahedrons are chosen is
crucial for the homogeneous method, where endo-orientated tetrahedrons tend to underestimate the strain, and conversely, the strain
is overestimated with a choice of epicardial orientated tetrahedrons.
Both methods follow the analytical strain relatively well for the E11 ,
E22 , E12 -components. The choice of fourth corner of the tetrahedrons in the non-homogeneous method has no influence on the discrete
strain values for these components. For the three remaining components, involving the radial (X3 ) direction, the homogeneous tetrahedron method deliver a large spread of estimated strains depending
on the choice of tetrahedrons. The maximum errors are substantially
greater in the homogeneous computations than the maximum errors
for the polynomial method.
Absolute errors for the polynomial method are showed in table 5.2,
with a mean error of all components of 0.017 ± 0.011. This method
is most accurate for E11 - and E12 -components where = 0.005. Table 5.1 displays the absolute errors when the homogeneous tetrahedron
method is applied on the cylinder model. Since there are six variants of
this method available, mean errors for each strain component are taken
for the three variants when endo-orientated tetrahedrons are chosen,
likewise for the variants with epi-orientated tetrahedrons. Mean component errors for these variants are 0.032 ± 0.013 and 0.025 ± 0.018
respectively. The tetrahedron method gives largest errors in the E33 component for both method variants, and when using the polynomial
method the largest error is found in the E13 -component.
5.6 Conclusion of the Comparison
29
Figure 5.3. Transmural strain components in a local cartesian coordinate
system (X1 , X2 , X3 ), for an analytical test case with a simulated 3-by-4
bead array. Solid line: analytical strain. Dotted line: polynomial strain.
Circles: epi-oriented tetrahedron strains. Crosses: endo-oriented strains.
5.6
Conclusion of the Comparison
When applied on a test cylinder model for systolic deformations, and
compared to an analytical cylinder strain, the polynomial strain estimation yields a higher accuracy, on the basis of absolute errors, than
the homogeneous tetrahedron estimation. For this situation, the absolute error is a good measure of accuracy since all the strain components
are relatively small and within the same magnitude. An alternative
would be relative error calculations, though that would be misleading
when dealing with strains close to zero, which is the case here.
The polynomial method also gives more stable and homogeneous results for all strain components, noticeable in the smaller standard
deviation. Important to remember is that these comparison results
come from a model simulation and do not reflect any measurement
30
Comparison of Strain Computation Methods
Table 5.1. Absolute strain errors , for the analytical test case with a
simulated 3-by-4 bead array. Homogeneous tetrahedron method used, mean
strain values and standard deviation showed for the three types of endooriented tetrahedrons available, likewise for the epi-oriented case.
Strain component
(E11 )
(E22 )
(E33 )
(E12 )
(E13 )
(E23 )
Mean ± SD
Endo oriented
0.019
0.026
0.052
0.004
0.033
0.034
0.032 ± 0.013
Epi oriented
0.016
0.010
0.058
0.030
0.026
0.033
0.025 ± 0.018
Table 5.2. Absolute strain errors , for an analytical test case with a
simulated 3-by-4 bead array. Mean value and standard deviation from all
components showed. Non-homogeneous polynomial fitting method (poly)
used.
Strain component
Poly
(E11 )
0.005
(E22 )
0.016
(E33 )
0.017
(E12 )
0.005
(E13 )
0.035
(E23 )
0.022
Mean ± SD
0.017 ± 0.011
errors but rather the inaccuracies of the analysis itself. The main difference between these strain estimation models is that the polynomial
method does not include any homogeneity assumption. Hence, errors
due to non-homogeneous deformations which are strongly present in
the ventricular wall are reduced [16]. The continuous describing of a
deformation that the polynomial method does enables the possibility
to calculate strain for as many arbitrary transmural points as wanted,
and to interpolate strain within the bead array volume [16].
Including measurement errors, the polynomial method has advantages
since it uses least squares approximations with spatial information
5.6 Conclusion of the Comparison
31
from all the beads, which minimizes the effects of random measurement errors in real data [16]. Whereas the homogeneous method only
makes use of four beads per strain calculation. Also, the stability
against missing markers has connections in number of beads involved
in the strain computation. Since there are 12 available markers in the
bead array and there are nine equations to be solved for the polynomial fitting method, a missing marker or two does not give any
noticeable effects on the resulting strain. For the homogeneous tetrahedron method, all six line segments forming a tetrahedron are needed
in order to solve the six strain components for that tetrahedron volume, though there are in total two markers in the bead array not
used. Moreover, an even better fitting of the polynomial estimated
strain could be obtained in the analytical test case, at least for the
E33 component, if a polynomial of higher order was used. Though
this would imply more noise sensitive results when applied on real
data sets, since all twelve beads have to be used to solve the twelve
cubic equations [13].
32
Comparison of Strain Computation Methods
Chapter 6
Method
The non-homogeneous polynomial fitting method will now be applied
on real data sets to compute the strains in the cardiac wall at different
wall depths and time points. By defining different coordinate systems,
strains referring to the orientation of the whole heart (cardiac strains)
or to the organization of the cardiac muscle fibers (fiber strains) can
be computed. Moreover, a relation between the two strain types is obtained by composing the cardiac strains into fiber strain components.
6.1
Coordinate Systems
The coordinates of the implanted radiopaque marker are measured in
an exterior, laboratory reference coordinate system. But in order to
receive relevant strains, that is to say strains that do not depend on
rigid body motions of the heart, a suitable reference coordinate system
is needed. It can be designed in many ways as long as it follows the
heart’s rigid body motion and describes the mechanics of the heart
meaningfully. Here two different approaches on coordinate system
designs are explained, one cardiac coordinate system and one fiber
coordinate system. Both systems are orthogonal and local cartesian,
and both are defined on parameters that change with time, thus the
coordinate systems have to be updated every time frame.
33
34
Method
(a)
(b)
Figure 6.1. (a) Cardiac coordinate system seen from a bead array position.
XC circumferential direction, XL longitudinal and XR radial. (b)Fiber
coordinate system of one laminar fiber sheet from the myocardium. Xf is
along the fiber direction, Xs aligns the sheet and Xn is normal to the sheet
plane.
6.1.1
Cardiac Coordinate System
The cardiac coordinate system is denoted (XC , XL , XR ), where XC is
circumferential, XL is longitudinal and XR is the radial base vector.
The radial direction is transmural, and normal to the left ventricular
wall, pointing away from the center of the left ventricle, hence created
as a normal to a plane made up from markers 15, 19 and 23. The XC axis is tangential to the cardiac wall, directed anterior-to-posterior on
the lateral side. A normal to a plane established by XR and a vector
from marker 1 in the apex to the centroid of markers 4, 7, 10 and
13, will establish the circumferential direction. See figure 4.1(a) for
marker specification. The longitudinal direction is tangential to the
wall, perpendicular to both XC and XR , and positive in the direction
apex-to-base. Hence, the XC XL -plane is tangential to the epicardium
of the left ventricle, with its normal XR being a transmural vector.
6.1.2
Fiber Coordinate System
The fiber coordinate system, (Xf , Xs , Xn ), is created in accordance
with the myocardial fiber architecture, as in figure 6.1(b). Xf is the
direction of the fibers, Xs is orientated along the myocardial fiber sheet
6.1 Coordinate Systems
35
plane orthogonal to Xf , and Xn is the axis normal to the sheet plane.
The fiber angle α is the angle between the circumferential direction
XC and the fiber direction Xf , and thus it describes the degree of the
fiber helices. The sheet angle β is the angle between the Xs -axis and
the XR -axis. The two coordinate systems with the fiber and sheet can
be seen in figure 6.2.
6.1.3
Transformation
Since both the coordinate systems are cartesian and orthogonal, they
can be transformed into each other by rotations. If the (XC , XL , XR )system in figure 6.2 first is rotated the angle α around the XR -axis,
XC will be transformed into the final Xf -axis position, and the new
coordinate basis will be (Xf , XL0 , XR ). (Xf , Xs , Xn ) is then obtained
by a subsequent rotation around the Xf -axis until that there is an
angle β between the XR and the rotating XL0 -axis, thus until a rotation
of π2 − β about Xf has been performed.
Figure 6.2. Tranformation by α and π2 − β rotation between the cardiac coordinate system (XC , XL , XR ) and the fiber coordinate system
(Xf , Xs , Xn ). XL0 is perpendicular to Xf , lies in the XC − XL -plane and is
rotated into Xs .
The rotation can be described according to Euler’s rotation theorem
36
Method
with a total transformation matrix Rtot as
Rtot = Rβ Rα ,
(6.1)
where each rotation is given by the respective rotation matrices


cosα sinα 0
Rα =  −sinα cosα 0 
(6.2)
0
0
1
 


1
0
0
1
0
0
Rβ =  0 cos( π2 − β) sin( π2 − β)  =  0 sinβ cosβ 
0 −cosβ sinβ
0 −sin( π2 − β) cos( π2 − β)
(6.3)
yielding Rtot as


cosα
sinα
0
Rtot =  −sinαsinβ cosαsinβ cosβ 
(6.4)
sinαcosβ −cosαcosβ sinβ
A point (C, L, R)T in the cardiac coordinate system can be expressed
in fiber coordinates via the following equation
 


f
C
 s  = Rtot  L 
(6.5)
n
R
6.2
Cardiac Cycle Timing
Strain is a relative concept, in need of a reference to be meaningful.
In this study it is taken on the basis of the cardiac cycle, and since
the diastole phase is in focus here, the reference is taken as the starting point of diastole. The time of filling onset (t = 0) is defined as
the time point where the pressure of the left ventricle reaches 10% of
its maximum variation for that heart beat. As mentioned in chapter
2.1.3, this is the time when the bicuspid valve between left atria and
left ventricle opens and thus blood filling begins.
To receive relevant strain results and most importantly to be able
to compare strains from data sets with different parameters, specific
6.3 Cardiac Strain Computation
37
points for where strain always is to be computed for are necessary.
These cardiac points are settled to be the time points at end diastole
(ED) and at end of early filling (EOEF). ED is found by looking at
the volume curve of the left ventricle and finding the closest subsequent time point from t = 0, where there is a local volume maximum.
The volume is obtained by, for every time frame, creating a convex
hull around the subepicardial markers 1-13. By definition, the point
of EOEF is simply taken 100 ms after t = 0, i.e. 6 sample points after
t = 0 when the sample frequency is 60 Hz. EOEF represents the first
third of the diastolic phase.
Figure 6.3. Left ventricular pressure (dashed line) and volume (solid line)
for one representative heart with marked cardiac cycle timings. Diastolic
phase starts with filling onset at t = 0 and ends at ED (End Diastole). End
Of Early Filling (EOEF) is defined as 100ms after filling onset.
6.3
Cardiac Strain Computation
Strain tensors are computed for the data sets described in chapter 4.2
according to the non-homogeneous polynomial method thoroughly explained in chapter 5.3. As for the cylinder model used in chapter 5
there are 12 markers in the bead arrays of the real data sets whose
coordinates for the deformed state are estimated with the polynomial
in equation 5.11. The cardiac coordinates of the 12 beads are acquired
38
Method
for the reference time frame (t = 0), and the matrix M in 5.12 can
be formed. The true deformed bead coordinates are the cardiac coordinates of the beads taken for the time point for which the strain
is to be computed for, i.e. point of EOEF or ED. After least-squares
approximations that yield three sets of constants ci (one for each direction XC , XL , XR ), a deformation gradient tensor can be estimated
according to equation 5.15, using cardiac reference coordinates of a
chosen point within in the bead array. This point has here been chosen at three different wall depths along a radial line from the centroid
of markers 15, 19 and 23 on the epicardium, to that radial coordinate
that the bead marker closest to the endocardium has. One subepicardial point at a 20% depth along the radial line from the epicardial
centroid, one in the midwall and one subendocardial point at a depth
of 80% were chosen.
Hence a cardiac strain tensor estimate at a specific time point (EOEF
or ED) and at a specific transmural position (subepicardial, midwall
or subendocardial) can be derived, looking like


ECC ECL ECR
E cardiac =  ECL ELL ELR 
(6.6)
ECR ELR ERR
Strains computed in this cardiac coordinate system give for example
information about changes in the wall thickness and wall stretching in
the circumferential direction.
6.4
Fiber Strain Computation
For every point and time frame that a cardiac strain (Ecardiac ) has
been computed, a transformation into fiber strain (Ef iber ) is made via
the rotation matrix Rtot and the angle values of α and β at given
depths. The transformation follows the equation
E f iber = Rtot E cardiac RTtot ,
(6.7)
where the resulting fiber strain tensor with its components looks like:


Ef f Ef s Ef n
E f iber =  Ef s Ess Esn 
(6.8)
Ef n Esn Enn
6.5 Fiber Strain Contributions to Cardiac Strain Components
39
Strains computed in the fiber coordinate system reflect changes in
length of the fibers, changes in distance between the fibers and between the fiber sheets, and/or changes in the thickness of the sheets.
The fiber strain can of course only be calculated for those sheep where
the histological measurements of the fiber and sheet angles have successfully been achieved.
6.5
Fiber Strain Contributions to Cardiac Strain Components
An inversion of equation (6.7) gives
E cardiac = RTtot E f iber Rtot ,
(6.9)
and by developing this for each cardiac strain component, six expressions in terms of fiber strain components are obtained:
ECC =
Ef f cos2 α
−Ef s sin2αsinβ
+Ess sin2 αsin2 β
+Ef n sin2αcosβ
+Enn sin2 αcos2 β
−Esn sin2 αsin2β
ELL =
Ef f sin2 α
+Ef s sin2αsinβ
+Ess cos2 αsin2 β
−Ef n sin2αcosβ
+Enn cos2 αcos2 β
−Esn cos2 αsin2β
ERR =
Ess cos2 β
+Enn sin2 β
+Esn sin2β
ECL =
1
E sin2α
2 ff
Ef s cos2αsinβ
− 12 Ess sin2αsin2 β − 12 Enn sin2αcos2 β
1
Ef n cos2αcosβ
E sin2αsin2β
2 sn
ECR = − 21 Ess sinαsin2β + 12 Enn sinαsin2β
+Ef s cosαcosβ
+Ef n cosαsinβ
ELR =
Ef f 21 cosαsin2β
+Ef s sinαcosβ
− 12 Enn cosαsin2β
+Ef n sinαcosβ
+Esn sinαcos2β
−Esn cosαcos2β
(6.10)
In this study especially the circumferential and radial cardiac strains
will be analyzed, giving information about which fiber strain components that contribute to the strain in the circumferential and radial
directions and how they effect it. The terms in the expressions above
2
CC
will be denoted as EfCC
f (= Ef f cos α) for the Ef f -term in ECC , Ess
40
Method
(= Ess sin2 αsin2 β) means the Ess -term in ECC and so on. Remarkable is that the radial strain does not depend on the fiber strain Ef f ,
nor on the fiber angle α. The circumferential strain has contributions
from each fiber strain component in its expression, while the radial
strain is solely composed of the normal strains in the sheet and the
normal directions, and shear strain in the plane of sheet-normal. The
normal fiber strains Ef f , Ess and Enn contribute to the normal cardiac strains ECC , ELL and ERR as their signs indicate, because the
expressions have no negative signs in front of these fiber strain nor
will the coupled trigonometric functions deliver negative outcome.
6.6
Statistics
Since the strain computation methods are applied on different animals
with different conditions, at two different sites of the ventricle, at three
transmural depths and at two time points, a large amount of data will
be obtained which can be analyzed in many ways. Restrictions on
what effects to be analyzed must be made, and statistical analysis
must be performed. An investigation of possible statistical models
suitable for this type of myocardial strain tensor study has been done
in [12], from where the models used here are assumed. The analysis
here are made in two parts, considered separately. In a first part
focusing on the effects of depth, site and time within baseline tissue.
Then including results from ischemic tissue in a second part, where
ischemic strains are compared to baseline strains at corresponding
depth, site and time points. No other effects than that of tissue type
were here considered.
6.6.1
Mean Values and Standard Errors
Two sets of strain components (one at ED and one at EOEF) are
calculated for each heart beat available for each animal. Predominantly are three-beat averages used to characterize the strain for each
animal and site (in some cases were only two heart beats available).
Mean strain values are calculated for all animals within one group,
and showed together with standard errors (as Mean ±SE)
6.6 Statistics
6.6.2
41
Significance Tests
When examining what effects the time has on the strains (cardiac
strain as well as fiber strain), other effects like transmural depth and
ventricular site are ignored. Hence, the wall depth and ventricular
site, and of course type of tissue (baseline or ischemic), are kept constant while comparing strains from different time points. The strain is
here defined to be zero at filling onset (t = 0).Tto evaluate the effects
of time, statistical t-tests are made to see if the strain components
at EOEF and ED are significantly deviated from zero, respectively.
This test is a one-sample t-test with the null hypothesis that the data
come from a distribution with mean zero, i.e. that the strains at t = 0
and EOEF (or ED) are the same. The significance level of the test is
the probability p to find values that implicates a rejection of the null
hypothesis even though it is true. Accepted level of significance where
the null hypothesis can be rejected is p < 0.05. The data are assumed
to come from a normal distribution with unknown variance. [15, 12]
Moreover for analyzing time effects, comparisons between the computed strains at EOEF and at ED are statistically evaluated with a
paired t-test. The data compared come from two dependent populations, meaning the strains at EOEF and ED are taken from the same
animal. The null hypothesis is here that the paired data come from
distributions with equal means. Accepted level of significance is set
to p < 0.01, hence the simultaneous acceptance level when each strain
component tested at all three depths will add up to p < 0.03. The
difference of the data sets are assumed to come from a normal distribution with unknown variance.
For the second part of the analysis, regarding baseline and ischemic
tissue comparison, unpaired t-tests are performed on data from different tissue types but at the same site, time and wall depth. In an
unpaired t-test the two data sets are assumed to be independent and
to origin from different normal distributions with unknown but equal
deviations. No significance tests of time were performed in this part
of analysis.
42
Method
6.7
Missing Markers
Sometimes during the invasive marker technique will an implanted
marker be dislocated for some reason, and its coordinates not acquired
correctly. Missing markers among the markers outlining the left ventricle (i.e. markers 1-13), require compensations to reduce errors in the
final strains. Regarding volume calculation, possible missing markers
are simply ignored. Though when defining the cardiac coordinate system it is crucial if any of the markers in the basal plane is missing.
The vector formed from the apex (marker 1) to the centroid of the
markers in the basal plane (markers 4, 7, 10 and 13), is defined as
a vector from markers 1 to a centroid of those markers in the basal
plane which are not missing.1
1
Marker 1 is not a missing marker in any of the ovine data files.
Chapter 7
Results
The first part of the result chapter will focus on the baseline data
derived from healthy sheep and its effects of time. Cardiac and fiber
strains at three transmural depths are presented and results from one
lateral and one anterior cardiac site are compared. Further, in a second part, equivalent results from ishemic tissue are presented and
compared to those from the baseline data. All resulting strain values
are specified in tables in the appendix.
7.1
7.1.1
Baseline
Cardiac Strains
Cardiac strains of sites and time points for baseline tissue are presented
in table A.2 in the appendix and figure 7.1. Significances from performed t-test are marked as asterisks in both tables and figures. During early filling, circumferential strain on the lateral site reaches its end
diastolic strain value already in the early filling phase in the subepicardium (ECC = 0.07±0.01). In figure 7.2(a), where the cirumferential
strain is plotted against percentage filling volume of the left ventricle, the subendocardial ECC increases almost linearly. At the anterior
site the circumferential strain is about two thirds of the total diastolic
strain, by the time of end of early filling (EOEF), for all wall depths
(71%, 76% and 78% for subepi, mid and subendo respectively). All
circumferential strains, for both the sites and all wall depths througout
diastole, are significant different from zero. The radial strains on the
43
44
Results
lateral side has a linear appearance in figure 7.2(b). However on the
anterior side, there are significant decreases in radial strains already at
EOEF for all depths (ERR = −0.13 ± 0.03; −0.16 ± 0.02; −0.16 ± 0.04),
where two thirds of the total diastole strain has been achieved.
Significant effects of time were only achieved for the subepicardial
circumferential-radial shear component, in addition to the ECC strains,
on the lateral side. However, on the anterior site almost all subepicardial cardiac strain components had changed significantly from zero
(p < 0.06 for ECL ), as well as all midwall components except ECR .
On the anterior endocardial side the normal strains dominated .
At ED, circumferential strain and longitudinal-radial shear strain dominated at all transmural depths for the lateral side (ECC = 0.07 ±
0.01; 0.08±0.02; 0.11±0.03 and ELR = −0.1±0.02; −0.1±0.02; −0.1±
0.03), together with longitudinal and radial normal strains at midwall
and subendocardial. The anterior side had about the same dominating strain components during all through diastole, namely the normal
strains.
Most significant changes between EOEF and ED are for the lateral
site found in the midwall (ELL , ECL and ELR ) and subendocardium
(ECC , ELL and ELR ), in contrast to the anterior site where most significant changes occure in subepicardium (ECC , ELL and ERR ) and
midwall (ELL and ERR ).
7.1.2
Fiber Strains
The only fiber strain component that changed significantly during
early filling were the subepicardial fiber lengthening and midwall fibersheet shear strain (Ef f = 0.03 ± 0.01; Ef s = −0.04 ± 0.01) for the
lateral side. Whilst the anterior side had significances in midwall and
subendocardial fiber strain (Ef f ), possibly also for the subepicardium
where p < 0.07, and furthermore in subepicardial fiber-normal shear
strain, midwall sheet strain and sheet-normal shear, and subendocardial normal strain. In table A.2 are all fiber strains from baseline data
presented.
7.1 Baseline
45
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.1. Bar plots over cardiac strain components (mean value ± SE)
for reference tissue, at the time points of end of early filling (EOEF) and
end diastole (ED), at lateral and anterior sites for the 3-by-4 bead array and
at three transmural depths; subepicardium, midwall and subendocardium.
*p<0.05, for a one-sample t-test comparing to zero. **p<0.01, for a paired
t-test comparing EOEF to ED. (a) ECC , (b) ELL , (c) ERR , (d) ECL , (e)
ECR , (f) ELR
46
Results
(a)
(b)
Figure 7.2. Lateral (circles) and anterior (triangles) strains for the
subepicardium (open symbols) and subendocardium (filled symbols) plotted against percentage filling volume of the left ventricle. (a) Cicrumferential strain, ECC . (b) Radial strain, ERR . Percentage filling volume at end
of early filling (EOEF) for lateral site (40%) and anterior site (48%) are
marked (The mean left ventricular volumes are different for the two sites,
because not all of the sheep received bead arrays at both sites.).
7.1 Baseline
47
At end diastole, the fiber strain (Ef f ) had increased significantly at all
depths at both the lateral (Ef f = 0.07 ± 0.02; 0.09 ± 0.03; 0.11 ± 0.03)
and the anterior site (Ef f = 0.12 ± 0.03; 0.15 ± 0.04; 0.19 ± 0.03).
The sheet strain had, on the lateral site, decreased significantly at
subepicardium and increased almost significantly (p < 0.06) at midwall, while a significant decrease can be seen for the anterior midwall. The sheet-normal shear strain was the dominating fiber strain
at midwall and subendocardial layers on both sites (lateral Esn =
0.11 ± 0.03; −0.17 ± 0.04 and anterior Esn = −19 ± 0.02; 0.16 ± 0.08)
at ED. The sheet strain (Ess ) and the sheet-normal strain (Esn ) had
consistently the opposite sign to that of the β-angle at corresponding
depth and site, with the exception of the sheet strain in the lateral
subendocardium at end diastole. See table 4.1 for fiber and sheet angles.
Only one strain component changed significantly during late filling on
the anterior side (midwall (Esn ), and three on the lateral side (midwall
and supendocardial Ef f and Esn at subendocardium).
7.1.3
Fiber Strain Contributions to Circumferential and Radial Strain
The lateral circumferential strain at EOEF is presented in terms of
contributing fiber strain components in figure 7.3. In the subendocardium, the circumferential expansion at EOEF origins almost exclusively from fiber expansion (EfCC
f 101%), also in the midwall is the
circumferential strain dominated by fiber expansion (EfCC
f 76%). However, the subepicardial circumferential strain can be assigned to a complex mixture of mostly fiber expansion (EfCC
f 22%), fiber-sheet shear
CC
CC
CC
18%).
(Ef s 29%), fiber-normal shear (Ef n 20%) and sheet-normal shear (Esn
The anterior side shows the same tendencies, though a slightly more
dominance of fiber strain also in subepicardium where EfCC
f is 58% at
EOEF.
An equivalent decomposition has been done for the end diastolic radial strain for lateral and anterior sites, in figure 7.4. The radial strain
only has three fiber strain terms, where the sheet-normal shear is the
48
Results
Figure 7.3. Lateral ECC decomposition into contributing fiber strain
terms at EOEF for basline myocardium. Each fiber term’s percentage
contribution to total circumferential expansion (dashed bars) is marked.
CC
Eff-term is EfCC
f , Ess-term is Ess etc.
largest contributor to wall thinning for all depths and sites. This effect
can be coupled to the result of Esn exhibiting opposite sign compared
to the β angle for all depths and sites, and thereby consistently contributing to a negative ERR . There are fiber strains counteracting wall
thinning, for example the sheet strain has a negative effect on lateral
RR
− 37%) at midwall, the normal strain counteracts
wall thinning (Ess
RR
wall thinning in the anterior midwall (Enn
− 13%).
7.2
Ischemic
The same methods have been applied on the ischemic data sets as
those for the baseline, and the results from the two data types are
compared in this chapter. Tables A.4 and A.5 presents the cardiac and
fiber strains repectively, with significant differences between baseline
and ischemic tissue marked.
7.2 Ischemic
(a)
49
(b)
Figure 7.4. Lateral (a) and anterior (b) ERR decomposition into contributing fiber strain terms at end diastole for baseline myocardium. Each
fiber term’s percentage contribution to total wall thinning (dashed bars) is
RR
marked. Eff-term is EfRR
f , Ess-term is Ess etc.
7.2.1
Cardiac Strains
Regarding cardiac strains, there are significant differences in the subendocardium in radial strain, circumferential-longitudinal shear and longitudinalradial shear at end diastole for the lateral site. The lateral site is the
region adjacent to where the coronary artery occlusion was provoked.
ELR at ED had changed significantly between the tissue types in the
subendocardium for both sites (lateral subendocardium: −0.1 ± 0.03
versus −0.21 ± 0.04, and anterior subendocardium: −0.05 ± 0.03 versus −0.15 ± 0.04). Possible significant changes of ELR are found in the
midwall at ED1 as well , indicating larger negative longitudinal-radial
strain for ischemic data. The anterior side, which is remote from the
infarction area, shows a reduction of the wall thinning at midwall in
ischemic tissue, where ERR has changed significantly throughout diastole (EOEF: −0.16±0.02 for baseline versus −0.05±0.04 for ischemic;
ED: −0.20±0.02 versus −0.07±0.05). The cardiac strains of ischemic
tissue are also presented as bar graphs in figure 7.5.
1
p<0.11 at lateral midwall and p<0.06 at anterior midwall
50
7.2.2
Results
Fiber Strains
Also the fiber strains show significant changes in some components
between healthy normal cardiac tissue and ischemic tissue. On the lateral side the shear strains were effected; subepicardial Ef n at EOEF
and Ef s at ED (0.04 ± 0.02 for baseline versus −0.02 ± 0.02 for ischemic), as well as subendocardial Esn at ED ( −0.17 ± 0.04 for baseline versus −0.02 ± 0.05 for ischemic) were significantly changed into
smaller absolute strain values for the ischemic data. On the anterior
side the fiber strain was significantly reduced at EOEF in the subendocardium (0.15 ± 0.02 for baseline versus 0.05 ± 0.02 for ischemic).
Sheet-normal shear was also significantly changed in midwall tissue,
from −0.13 ± 0.01 and −0.19 ± 0.02 in baseline tissue at EOEF and
ED respectively, into the positive values of 0.03 ± 0.04 and 0.03 ± 0.04
in ischemic condition.
7.2.3
Fiber Strain Contributions to Circumferential and Radial Strain
When focusing on how the circumferential strain is composed of fiber
strain terms for tissue next to the occlusion (lateral side), the fiber
strain contribution (EfCC
f ) dominates throughout the wall for the ischemic tissue. Figure 7.6 shows the lateral ECC decomposition at
EOEF.
The dominating sheet-normal shear contributor to radial strain in
baseline tissue is no longer as clearly present in the ischemic tissue.
RR
Looking at lateral ischemic tissue at ED, Esn
has been replaced by
RR
Enn , a normal strain contribution, at midwall and subendocardium,
as seen figure 7.7(a). Wall thinning in anterior midwall is mainly
RR
caused by sheet strain (Ess
100%) in ischemic data, and as figure
7.7(b) shows, the normal strain is here counteracting wall thinning
CC
− 51%). Additional counteraction is present in the subendo(Enn
cardium for both anterior and lateral walls, caused by sheet extension.
7.2 Ischemic
51
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.5. Bar plots over cardiac strain components (mean value ± SE)
for ischemic tissue, at the time points of end of early filling (EOEF) and end
diastole (ED), at lateral and anterior sites for the 3-by-4 bead array and
at three transmural depths; subepicardium, midwall and subendocardium.
*p<0.05, for a one-sample t-test comparing to zero. Statistically significant
difference from corresponding baseline cardiac strain, ++ p<0.05. (a) ECC ,
(b) ELL , (c) ERR , (d) ECL , (e) ECR , (f) ELR
52
Results
Figure 7.6. Lateral ECC composition into contributing fiber strain terms
at end of early filling for ischemic myocardium. Each fiber term’s percentage contribution to total circumferential expansion (dashed bars) is marked.
CC
Eff-term is EfCC
f , Ess-term is Ess etc.
(a)
(b)
Figure 7.7. Lateral (a) and anterior (b) ERR composition into contributing fiber strain terms at end diastole for ischemic myocardium. Each
fiber term’s percentage contribution to total wall thinning (dashed bars)
RR
is marked. Eff-term is EfRR
f , Ess-term is Ess etc.
Chapter 8
Discussion
A majority of cardiac strain studies are done to examine the mechanisms of myocardial contraction. In this project, however, focus has
been on diastole and what mechanisms that allow such a rapid and
effective blood filling.
8.1
Baseline
Most of the circumferential expansion is achieved already during early
filling for both a lateral and an anterior myocardial region, with the
exception of lateral subendocardial ECC which has an almost linear
increase with filling volume. Ashikaga et al. studied cardiac and fiber
strains during diastolic filling for an anterior position of the left ventricular wall of dogs [1]. They computed strains with ED as reference
configuration, not filling onset as in this study. When comparing to
their results, strains from this study have been recomputed using ED
as reference. Ashikaga et al. found that the ECC increased almost
linearly with filling volume for the subepicardium. Two thirds of the
total diastolic subendocardial circumferential strain was achieved during the first third of diastole in their study, which corresponds to the
phase of early filling.
The rapid early filling phase was dominated by the circumferential
stretch throughout the wall together with fiber lengthening on the lateral site. The anterior wall had significant large cadiac normal strains
as well as midwall and subendocardial fiber expansion and negative
53
54
Discussion
sheet-normal shear strain. The major fiber extension on the anterior
site occurred during the early filling phase, in accordance with the
findings of Ashikaga et al. [1]. The lateral side showed in our study
a more linear increase of fiber strains with percentage filling volume
of the left ventricle. The radial strain in the anterior region decreased
79% and 76% of the total diastolic radial strain during early filling
for the subepicardium and the subendocardium respectively. The corresponding strains in the study of Ashikaga et al. decreased around
60% and 62% during the same period. The lateral side in this study
only showed a 33% subepicardial decrease in early filling.
Previous examination of cardiac and fiber strains during systole made
by Cheng et al. present results of uniformly positive sheet extension (Ess ) and sheet thickening (Enn ) at each wall depth [5]. The
reverse is not true for diastolic fiber strains according to this investigation, where both sheet strain and normal-to-sheet strain change
signs throughout the wall. Ess behave like the sheet-normal shear
(Esn ) and assumes negative strain values when the sheet angle β is
positive. Cheng et al. observed the reverse behaviour for systole,
where Esn exhibited the same sign as β [5]. Ashikaga et al. [1] looked
at only two transmural points; subepicardium and subendocardium.
They pointed out a transmural gradient of Esn , being largest at the
subendocardium (0.167 vs. 0.067 at filling onset). This statement
and these strain values are in accordance with the findings of lateral
strains from this study. Here, the lateral sheet-normal shears, recomputed with ED as reference, are 0.19 and 0.07 for the subendocardium
and subepicardium, respectively. However, the anterior site had in
this study negative Esn (−0.22 at subendocardium versus −0.09 at
subepicardium), though diminishing strain from subendocardium to
subepicardium.
Notable is that since also the midwall sheet-normal shear was computed here, yeilding strains of opposite signs to those of the outer
and inner wall parts, discussions about a transmural Esn gradient
should be done carefully. Moreover, since the β-angle seems to have
impact on Esn , the angles need to be included in this discussion of
comparison with Ashikaga et al. The fiber angle had similar behaviour in the study of Ashikaga et al. [1] (α ≈ −15◦ ; 30◦ ; 70◦ at
8.2 Ischemic
55
subepicardium, midwall and subendocardium) as in this study, though
the values differ. However, the sheet angles reported by Ashikaga et
al. (β ≈ −40◦ ; −42◦ ; −20◦ at subepicardium, midwall and subendocardium) does not match any sheet angle behaviour from either myocardial region studied here.
Wall thinning during diastole is showed here to mainly be caused by
the negative shearing of myocyte fibers in the sheet-normal direction,
meaning a sliding of the fiber sheets in the sheet-normal plane. Previous systolic studies, like [5] of Cheng et al., also showed a substantial
RR
contribution of sheet-normal shear to wall thickening (Esn
), as well
RR
)
as consistent positive contributions of sheet thickening (positive Enn
RR
and sheet extension (positive Ess ) for the ovine hearts. However, here
the normal and sheet strains counteract the diastolic wall thinning in
subepicardium and midwall respectively.
Although no statistical tests have here been made to compare strains
at different wall depth and at different sites, remarks like that the anterior site has consistently larger absolute mean cardiac strain values,
as well as fiber strain values, than the lateral strains at corresponding depth and time (except for the longitudinal-radial shear) can be
made. Moreover, the circumferential, longitudinal, radial and fiber
strains have larger absolute mean values in the subendocardium than
the midwall and smallest values in the subepicardium.
8.2
Ischemic
An understanding of the mechanical alterations in ischemic cardiac
tissue could be important in the design of future surgical therapy to
prevent left ventricular remodeling. For example an insertion of a
Cardiac Support Device has been showed by Cheng et al. in [6] to
attenuate infarcted induced shear strain abnormalities.
An investigation of how a localized infarction perturbs transmural
strain patterns during systole in regions adjacent to and remote from
the infarction, have been performed by Cheng et al. [3]. They showed
that transmural shear strains increased not only in adjacent regions
56
Discussion
but also at a site remote from a localized infarction. In particular was
an increase of the positive longitudinal-radial shear observed in the
inner layers of the myocardium, lateral and anterior. Also in diastole
the londitudinal-radial shear seems to be effected when infarction is
provoked. According to the results of this study, ELR assumes even
larger negative strain values for the ischemic tissue in both lateral and
anterior sites.
It has been proposed that the increased wall strain/stress adjacent
to the infarcted area is the reason to why regional dysfunction from a
localized infarction can spread to effect the whole ventricle [3]. Altered
strain patterns will in turn evoke cytokine and reactive oxygen species
production, which stimulate myocyte apoptosis, extracellular matrix
disruption and fibrosis [3]. This means wall thinning and stretching,
and thus further exaggerated wall stress leading to a positive feedback loop and global cardiac dysfunction. In this study, the fiber
strain (Ef f ) on the anterior side in ischemic tissue was reduced compared to control conditions, while the lateral side showed consistently
larger mean fiber strain (Ef f ) values. This means that on the anterior
site, which is remote from the ischemic area, the fibers expand less
than normal, becomes stiffer, whereas the adjacent to the ischemic
area is subjected to larger fiber stretches than normal. This could
support the above theory of wall thinning and stretching in adjacent
regions spreading dysfunction like the perturbations of longitudinalradial shears.
In normal tissue the circumferential subepicardial strain is divided
in contributions from almost all fiber strain components. This complex mixture is no longer seen in the ischemic case. Fiber elongation
(Ef f ) is almost exclusively responsible for the circumferential expansion throughout the cardiac wall. This implicates that the fiber strains
have been altered by the ischemic condition in such a way that the
normal mechanisms allowing effective filling no longer work and thus
other mechanisms compensates. For example were both the fibernormal shear (Ef n ) and the fiber-sheet shear (Ef s ) altered in the
subepicardium in the ischemic data, which are the two largest contributors to circumferential expansion in control conditions.
8.3 Conclusions
57
Also the wall thinning mechanism in diastole seems to be altered in
ischemic myocardium. For both lateral and anterior sites are the midwall and subendocardial wall thinning composed differently than in
baseline myocardium. The dominating contribution of sheet-normal
shear in baseline midwall, has been replaced by sheet thinning (negative Enn ) and sheet shortening (negative Ess ) in lateral and anterior positions, respectively. Neither sheet strain nor normal strain
were close to be significantly changed in the comparison of baseline/ischemic fiber strains, see table A.4. However, sheet-normal shear
was seriously altered in ischemic tissue with significant decreases in
midwall and subendocardium. Remarkable is that even though the
mean Esn values in lateral subepicardium has changed from being
small but negative in baseline into larger positive values in ischemia,
and thereby not following the coupling to opposite sign of β, Esn still
constitutes 69% of the wall thinning in that position. The larger counteraction from sheet strain in lateral subendocardium and from normal strain in anterior midwall is reflected in the significant decreases
in wall thinning (i.e. less negative radial strains) for these positions,
seen in table A.4.
8.3
Conclusions
In conclusion, transmural cardiac and fiber strain during diastole have
complex appearance with regional and temporal variations. Most of
the circumferential expansion obtained during diastole, is achieved already during early filling for both a lateral and an anterior myocardial
region of healthy sheep hearts. The wall thinning process during diastole in normal cardiac tissue is mainly caused by a shearing mechanism
in a sheet-normal plane. An ischemic condition involve significant alterations of longitudinal-radial shear strain. The complex mixture
of fiber strains contributions to epicardial circumferential expansion,
seen in basline, has been replaced by a dominating fiber elongation
contribution in ischemic tissue. Also the wall thinning mechanisms in
diastole are altered in ischemic myocardium.
58
Discussion
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Appendix A
Tables
Table A.1. List over experimental sheep manipulations, their correspondong site of bead array insertion; lateral and/or anterior. ? : the fiber and
sheet angles (α and β) have successfully been measured (within paranthesis
is stated for which site the angles have been measured, if the animal has
bead arrays on both sites). Different sheep were used for baseline study
and ischmic tissue study.
Baseline (N = 15)
Ischemic (N = 10)
Animal no
Site
α, β
Animal no
Site
α, β
B1
lat
?
I1
lat
I2
lat+ant ? (lat+ant)
B2
lat
?
I3
lat, ant ? (lat+ant)
B3
lat
?
B4
lat
?
I4
lat
?
B5
lat
?
I5
lat
?
B6
lat
?
I6
lat+ant ? (lat+ant)
B7
lat
?
I7
lat+ant ? (lat+ant)
I8
lat+ant ? (lat+ant)
B8
ant
?
B9
lat
I9
lat
?
I10
ant
?
B10
lat+ant ? (ant)
B11
lat+ant ? (ant)
B12
ant
?
B13
ant
B14
ant
?
B15
ant
?
63
64
Tables
Table A.2. Cardiac strains of baseline tissue at three transmural depths
for time points of end of early filling (EOEF) and end diastole (ED). Strains
from 3-by-4 bead arrays in both a lateral and an anterior position in the left
ovine ventricle are listed. Statistically significant effects of time: *p<0.05:
compared to reference time, †p<0.01: EOEF compared to ED. Mean ± SE
strain values from N animals.
EOEF
ED
Lateral site (N=10)
EOEF
ED
Anterior site (N=7)
Subepicardium
ECC 0.07 ± 0.01*
ELL
0.01 ± 0.02
ERR −0.01 ± 0.03
ECL
0.02 ± 0.01
ECR −0.04 ± 0.01*
ELR −0.03 ± 0.02
0.07 ± 0.01*
0.04 ± 0.02
−0.05 ± 0.03
−0.01 ± 0.01
−0.02 ± 0.02
−0.1 ± 0.02*
0.10 ± 0.04*
0.14 ± 0.03*†
0.09 ± 0.03*
0.15 ± 0.03*†
−0.13 ± 0.03* −0.17 ± 0.03*†
0.03 ± 0.01
0.04 ± 0.01*
−0.08 ± 0.02* −0.05 ± 0.03
−0.07 ± 0.02* −0.06 ± 0.02*
Midwall
ECC 0.06 ± 0.02*
ELL
0.02 ± 0.03
ERR −0.04 ± 0.03
ECL
0.03 ± 0.01
ECR −0.02 ± 0.02
ELR −0.03 ± 0.02
0.08 ± 0.02*
0.08 ± 0.03*†
−0.09 ± 0.02*
0.00 ± 0.01†
−0.01 ± 0.02
−0.1 ± 0.02*†
0.13 ± 0.04*
0.17 ± 0.04*
0.10 ± 0.02*
0.18 ± 0.03*†
−0.16 ± 0.02* −0.20 ± 0.02*†
0.05 ± 0.02*
0.06 ± 0.03
−0.02 ± 0.02
−0.01 ± 0.02
−0.05 ± 0.02* −0.05 ± 0.02*
Subendocardium
ECC 0.05 ± 0.02*
ELL
0.03 ± 0.03
ERR −0.06 ± 0.03
ECL
0.01 ± 0.01
ECR −0.01 ± 0.03
ELR −0.03 ± 0.03
0.12 ± 0.03*†
0.15 ± 0.05*†
−0.11 ± 0.02*
−0.00 ± 0.02
0.01 ± 0.03
−0.1 ± 0.03*†
0.14 ± 0.02*
0.11 ± 0.04*
−0.16 ± 0.04*
0.04 ± 0.02
0.05 ± 0.03
−0.03 ± 0.02
0.18 ± 0.03*
0.21 ± 0.04*†
−0.21 ± 0.03*
0.03 ± 0.04
0.05 ± 0.03
−0.05 ± 0.03
65
Table A.3. Fiber strains of baseline tissue at three transmural depths for
time points of end of early filling (EOEF) and end diastole (ED). Strains
from 3-by-4 bead arrays in both a lateral and an anterior position in the left
ovine ventricle are listed. Statistically significant effects of time: *p<0.05:
compared to reference time, †p<0.01: EOEF compared to ED. Mean ± SE
strain values from N animals.
EOEF
ED
Lateral site (N=7)
EOEF
ED
Anterior site (N=6)
Subepicardium
Ef f
0.03 ± 0.01*
Ess −0.01 ± 0.02
Enn
0.05 ± 0.02
Ef s
0.02 ± 0.02
Ef n −0.03 ± 0.01
Esn −0.01 ± 0.03
0.07 ± 0.02*
−0.06 ± 0.02*
0.05 ± 0.03
0.04 ± 0.02
0.03 ± 0.03
−0.03 ± 0.03
0.08 ± 0.04
0.02 ± 0.05
−0.03 ± 0.06
−0.02 ± 0.04
−0.05 ± 0.02*
0.03 ± 0.05
Midwall
Ef f
0.05 ± 0.03
Ess
0.00 ± 0.02
Enn −0.03 ± 0.03
Ef s −0.04 ± 0.01*
Ef n −0.02 ± 0.02
Esn
0.04 ± 0.03
0.09 ± 0.0*†
0.05 ± 0.02
−0.07 ± 0.03*
0.01 ± 0.02
−0.01 ± 0.02
0.11 ± 0.03*
0.12 ± 0.05*
0.15 ± 0.04*
−0.10 ± 0.02* −0.09 ± 0.03*
0.05 ± 0.03
0.09 ± 0.04
0.04 ± 0.02
0.04 ± 0.02
−0.05 ± 0.03
−0.05 ± 0.04
−0.13 ± 0.01* −0.19 ± 0.02*†
Subendocardium
Ef f
0.04 ± 0.03
Ess −0.00 ± 0.03
Enn −0.03 ± 0.04
Ef s −0.01 ± 0.03
Ef n −0.01 ± 0.03
Esn −0.05 ± 0.03
0.11 ± 0.03*†
0.02 ± 0.03
0.02 ± 0.05
−0.01 ± 0.02
−0.01 ± 0.04
−0.17 ± 0.04*†
0.15 ± 0.02*
0.01 ± 0.05
−0.07 ± 0.02*
0.01 ± 0.02
−0.05 ± 0.02
0.11 ± 0.05
0.12 ± 0.03*
0.00 ± 0.05
0.00 ± 0.07
−0.01 ± 0.05
−0.05 ± 0.01*
0.04 ± 0.07
0.19 ± 0.03*
0.04 ± 0.06
−0.04 ± 0.04
−0.02 ± 0.04
−0.06 ± 0.02*
0.16 ± 0.08
66
Tables
Table A.4. Cardiac strains of ischemic tissue at three transmural depths
for time points of end of early filling (EOEF) and end diastole (ED). Strains
from 3-by-4 bead arrays in both a lateral and an anterior position in the left
ovine ventricle are listed. Statistically significant effects of time: *p<0.05:
compared to reference time, statistically significant difference from corresponding baseline cardiac strain, ‡ p<0.05. Mean ± SE strain values from
N animals.
EOEF
ED
Lateral site (N=9)
EOEF
ED
Anterior site (N=6)
Subepicardium
ECC
0.06 ± 0.03
ELL
0.02 ± 0.03
ERR −0.01 ± 0.03
ECL −0.01 ± 0.01
ECR −0.03 ± 0.01*
ELR −0.06 ± 0.03
0.09 ± 0.03*
0.05 ± 0.03
−0.02 ± 0.06
−0.02 ± 0.02
−0.06 ± 0.02*
−0.12 ± 0.03*
0.05 ± 0.02*
0.04 ± 0.03
−0.06 ± 0.03
0.00 ± 0.01
−0.02 ± 0.03
−0.06 ± 0.02*
0.10 ± 0.03*
0.06 ± 0.04
−0.08 ± 0.03
0.01 ± 0.02
−0.01 ± 0.03
−0.08 ± 0.03*
Midwall
ECC
0.05 ± 0.02
ELL
0.03 ± 0.04
ERR −0.02 ± 0.01
ECL −0.02 ± 0.02‡
ECR −0.03 ± 0.01*
ELR −0.08 ± 0.03*
0.12 ± 0.03*
0.07 ± 0.04
−0.05 ± 0.03
−0.50 ± 0.02
−0.04 ± 0.02
−0.16 ± 0.03*
0.05 ± 0.01*
0.05 ± 0.05
−0.05 ± 0.04‡
−0.02 ± 0.01‡
0.01 ± 0.01
−0.07 ± 0.02*
0.13 ± 0.03*
0.10 ± 0.05
−0.07 ± 0.05‡
−0.00 ± 0.02
0.00 ± 0.02
−0.11 ± 0.03*
Subendocardium
ECC 0.06 ± 0.02*
ELL
0.04 ± 0.05
ERR
0.00 ± 0.03
ECL −0.04 ± 0.02
ECR −0.02 ± 0.03
ELR −0.1 ± 0.04*
0.16 ± 0.04*
0.11 ± 0.06
−0.03 ± 0.03‡
−0.09 ± 0.04*‡
0.00 ± 0.05
−0.21 ± 0.04*‡
0.10 ± 0.02*
0.16 ± 0.03*
0.04 ± 0.05
0.12 ± 0.05
−0.03 ± 0.07
−0.05 ± 0.08
0.03 ± 0.03
−0.02 ± 0.04
0.03 ± 0.02
0.02 ± 0.02
−0.09 ± 0.02* −0.16 ± 0.04*‡
67
Table A.5. Fiber strains of ischemic tissue at three transmural depths for
time points of end of early filling (EOEF) and end diastole (ED). Strains
from 3-by-4 bead arrays in both a lateral and an anterior position in the left
ovine ventricle are listed. Statistically significant effects of time: *p<0.05:
compared to reference time, statistically significant difference from corresponding baseline fiber strain: ‡ p<0.05. Mean ± SE strain values from N
animals.
EOEF
ED
Lateral site (N=8)
Subepicardium
Ef f
0.07 ± 0.03*
Ess −0.03 ± 0.03
Enn
0.02 ± 0.03
Ef s −0.01 ± 0.01
Ef n 0.01 ± 0.01‡
Esn
0.06 ± 0.03
EOEF
ED
Anterior site (N=6)
0.11 ± 0.01*
−0.06 ± 0.04
0.03 ± 0.04
−0.02 ± 0.02‡
−0.01 ± 0.02
0.03 ± 0.05
0.04 ± 0.01*
0.00 ± 0.02
−0.02 ± 0.03
0.01 ± 0.02
−0.01 ± 0.03
0.03 ± 0.03
0.08 ± 0.02*
0.01 ± 0.03
−0.03 ± 0.04
0.00 ± 0.03
−0.04 ± 0.03
0.05 ± 0.04
Midwall
Ef f
0.07 ± 0.03*
0.15 ± 0.03*
Ess
0.08 ± 0.05
0.12 ± 0.06
Enn −0.07 ± 0.03* −0.12 ± 0.04*
Ef s
0.01 ± 0.03
0.01 ± 0.04
Ef n
0.02 ± 0.01
0.02 ± 0.02
Esn
0.01 ± 0.01
0.04 ± 0.02
0.05 ± 0.01*
−0.07 ± 0.04
0.07 ± 0.04
0.00 ± 0.01
0.02 ± 0.01‡
0.03 ± 0.04‡
0.13 ± 0.03*
−0.10 ± 0.05
0.12 ± 0.05
0.01 ± 0.02
0.00 ± 0.01
0.03 ± 0.04‡
Subendocardium
Ef f
0.05 ± 0.03
Ess
0.09 ± 0.10
Enn −0.02 ± 0.03
Ef s −0.04 ± 0.03
Ef n
0.00 ± 0.01
Esn −0.01 ± 0.03
0.05 ± 0.02*‡
0.05 ± 0.06
0.00 ± 0.05
0.01 ± 0.02
0.03 ± 0.02‡
0.01 ± 0.06
0.14 ± 0.04
0.10 ± 0.09
−0.01 ± 0.07
−0.04 ± 0.03
0.04 ± 0.03‡
0.02 ± 0.07
0.14 ± 0.05*
0.18 ± 0.13
−0.04 ± 0.05
−0.02 ± 0.06
−0.01 ± 0.02
−0.02 ± 0.05‡
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